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Φ
be an EIM for the elements of which there exists an evaluating function
. Then
Φ(
a
) k 1 , l ρ( 1 ) Φ(
a
) k 2 , l ρ( 2 ) ...Φ(
a
) k m , l ρ( m ) ,
[ ρ(
1
),...,ρ(
m
) ]
if m
=
car d
(
K
)
car d
(
L
) =
n
per
Φ (
A
) =
,
Φ(
a
) k σ( 1 ) , l 1 Φ(
a
) k σ( 2 ) , l 2 ...Φ(
a
) k σ( n ) , l n ,
[ σ(
1
),...,σ (
n
) ]
=
(
)
(
) =
if m
car d
K
car d
L
n
where
ρ :{
1
,
2
,...,
m
}→{
1
,
2
,...,
n
}
is a bijection from
{
1
,
2
,...,
m
}
in
{
1
,
2
,...,
n
}
and
σ :{
1
,
2
,...,
n
}→{
1
,
2
,...,
m
}
is a bijection from
{
1
,
2
,...,
n
}
in
.
And yet, we can keep the concept of a determinant adding additional condition.
For brevity, here we discuss only the case of square EIM, i.e., the sets K and L have
equal number of elements. Let us fix the order of the elements of set K as a vector
[
{
1
,
2
,...,
m
}
k 1 ,...,
k m ]
. Each of its permutations
[
k ρ( 1 ) ,...,
k ρ( m ) ]
will be estimated as odd
or even about the vector
[
k 1 ,...,
k m ]
. Then, the determinant of the EIM A can be
defined by
det Φ, [ k 1 ,..., k m ] (
A
)
) [ k 1 ,..., k m ] Φ(
=
(
1
a
) k 1 , l ρ( 1 ) Φ(
a
) k 2 , l ρ( 2 ) ...Φ(
a
) k m , l ρ( m ) .
[ ρ(
1
),...,ρ(
m
) ]
3.7 Transposed EIM
Let the EIM A be given as above. Then its Transposed EIM has the form
k 1
...
k i
...
k m
l 1 a l 1 , k 1
...
a l 1 , k i
...
a l 1 , k m
.
.
. .
.
. . .
...
A =[
L
,
K
, {
a l j , k i }] =
.
l j a l j , k 1
...
a l j , k i
...
a l j , k n
.
.
. .
.
. . .
...
l n a l n , k 1
...
a l n , k i
...
a l n , k m
The geometrical interpretation is
 
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